A smartphone's battery percentage decreases at a constant rate while a video is playing. After 20 minutes, the battery reads...

1. TRANSLATE the problem information

• Given information:
  • Battery decreases at constant rate (linear relationship)
  • After 20 minutes: 88% battery
  • After 100 minutes: 56% battery
  • Need to find: battery percentage after 85 minutes

• What this tells us: We have two coordinate points (20, 88) and (100, 56) for a linear model

2. INFER the mathematical approach

• Since the rate is constant, we need a linear equation \(\mathrm{B = mt + b}\)
• We can find the slope using our two points, then use point-slope form
• The slope represents the rate of battery decrease per minute

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = (y_2 - y_1)/(x_2 - x_1)}\)
• \(\mathrm{m = (56 - 88)/(100 - 20)}\)
  \(\mathrm{m = -32/80}\)
  \(\mathrm{m = -0.4\% \text{ per minute}}\)
• The negative slope confirms the battery is decreasing

4. INFER the best equation form and set it up

• Point-slope form is efficient: \(\mathrm{B - y_1 = m(t - x_1)}\)
• Using point (20, 88): \(\mathrm{B - 88 = -0.4(t - 20)}\)
• Rearranging: \(\mathrm{B = 88 - 0.4(t - 20)}\)

5. SIMPLIFY by substituting \(\mathrm{t = 85}\)

• \(\mathrm{B = 88 - 0.4(85 - 20)}\)
• \(\mathrm{B = 88 - 0.4(65)}\)
• \(\mathrm{B = 88 - 26 = 62\%}\)

Answer: C. 62%

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that 'constant rate' means linear relationship, or they might confuse which values represent the coordinates.

Some students might try to find a pattern like '88 - 56 = 32, so the battery drops 32% every 80 minutes' without properly setting up the linear model. This oversimplified thinking doesn't account for the proportional relationship and leads to incorrect calculations.

This may lead them to select Choice A (58%) or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors, particularly when calculating the slope or during the final substitution.

Common calculation mistakes include: getting \(\mathrm{-32/80 = -0.5}\) instead of -0.4, or computing \(\mathrm{0.4 \times 65}\) incorrectly. Even small arithmetic errors compound to wrong final answers.

This may lead them to select Choice B (60%) or Choice D (64%).

The Bottom Line:

This problem tests whether students can recognize real-world linear relationships and translate them into mathematical models. The key insight is that 'constant rate' always signals linear behavior, and having two data points gives you everything needed to build that linear equation.